The root system generated by the \(v_1,\dots , v_{\nu (p-1)}\)  has only summands of type \(A\) , \(D\)  or \(E\)  and its Weyl group contains \(g_*\)  as an element of prime order \(p\) . R.W. Carter's Table 3 in [1]} tells us that such an element
is given by a Coxeter transformation of a root subsystem \(R_0\) , all of whose components
are of type \(A_{p-1}\) . We are given that \(\mathbf {L}_G\)  is contained in a \(\mathbb {Z}[G]\) -submodule of \(\mathbf {\Lambda }\)  that is isomorphic to \(\mathbb {Z}[G]^\nu \) . This means that there exist \(v_1, \dots , v_\mu \)  in \(\mathbf {\Lambda }\) 
such that \(L_G\)  has basis
\(\lbrace g^j(g-1)v_i\, :\, i=1, \dots , \nu ; j=1, \dots , p-1\rbrace .\) 
